When is a real-analytic function harmonic?

Question

I recently learnt that every harmonic function occurs as the real part of a complex analytic function. We also know that every harmonic function is real analytic. So, when is a real-analytic function harmonic ?

Answer 1

When $-\Delta u=-\sum_{k=1}^n\frac{\partial^2u}{\partial x_k^2}=0$

I believe an equivalent property is the mean integral property, I.e.:

$u(x)=\frac{1}{n\alpha(n)r^{n-1}}\int_{\partial B(x,r)}u(y)dS(y)=\frac{1}{\alpha(n)r^{n}}\int_{ B(x,r)}u(y)dy$, $\forall r\gt 0$.

Where $\alpha(n)$ is the volume on the unit sphere in $\Bbb R^n$.